Unraveling Linear Equations: Do More Variables Mean More Solutions?

When we think about systems of linear equations, a common assumption is that having more variables than equations always leads to infinitely many solutions. But is this always true? Let’s explore!

For example, let's consider two equations:

x + y + z = 1 [eqA]

3x + y + z = 1 [eqB]

From these equations, we conclude: z + y + z = 3x + y + z

\=> z = 3x [eq1] (subtracting y + z from both sides)

We know there are infinite solutions for this equation [eq1]. So, is it correct to say that more variables than equations always lead to infinite solutions? No, here is a counterexample:

x + y + z = 1 [eqC]

x + y + z = 2 [eqD]

In these equations, there are also more variables than equations, but no solution exists.

By Gaussian elimination:

Subtracting eqC from eqD, we get:

x + y + z =1

0 = 1

This is a contradiction, meaning the system is inconsistent. Even though there are more variables than equations, no solution exists.

Conclusion:

A system with more unknowns than equations may have:

1. Infinitely many solutions (this is the typical case if the system is consistent and underdetermined).

2. No solution, if the equations are inconsistent (as shown in the counterexample).

The takeaway? Having more variables than equations doesn’t guarantee infinite solutions—it depends on the consistency of the system.